Question 356729
Assuming addition of vectors and scalar multiplication in {{{R^n}}}, two conditions had to be met for a subset to be a subspace:
i)  If u and v are vectors in W, then u + v is in W, and
ii) If u is in W, and c is a a scalar, then c*u is in W.
For the question above, 
i) (x, y, x-y)+(z, w, z-w) = (x+z, y+w, x-y+z-w) = (x+z, y+w, (x+z)-(y+w)),
ii)c*(x, y, x-y) = (cx, cy, c(x-y)) = (cx, cy, cx-cy).
Therefore W is a subspace of {{{R^3}}}.